Density Gradient Scale Length Research Articles

A k-space integral equation for describing propagation through a strongly inhomogeneous plasma density profile

A k-space integral equation is derived that describes the propagation of electromagnetic waves induced by an external source of charge or current in a magnetized plasma (B=B0ẑ) having an arbitrary density variation in the x̂ direction. The nonlocal k-space dielectric tensor kernel is derived keeping finite ion Larmor radius ρi corrections to all orders without the use of an expansion in the inverse density gradient scale length LN so that the effect of a strongly inhomogeneous plasma density profile (LN≊ρi) on wave propagation in the ion cyclotron range of frequencies can be studied. The integral equation is solved numerically in the electrostatic limit to study the capacitive excitation of ion Bernstein waves for frequencies near the second harmonic of the ion cyclotron frequency (ω≊2Ωi). The spectrum of weakly damped eigenmodes for a plasma having a large region of uniform density and a highly nonuniform edge is found to consist of numerous ‘‘uniform plasma’’ modes and an electrostatic drift mode that propagates only in the edge region. Asymmetries in the radial structure of these modes, which arise from the diamagnetic drift of particles in the plasma edge, result in an asymmetric distribution of wave energy launched in the directions parallel and antiparallel to the diamagnetic current. The surface electrostatic drift mode is found to be the dominant mode of oscillation as the wave frequency approaches the second harmonic of the ion cyclotron frequency.

Reflectivity of steep-gradient plasmas in intense subpicosecond laser pulses.

The wave equation for laser light incident on a static, steep-gradient plasma has been solved numerically, with the inclusion of an intensity-dependent electron-ion collision rate, and incorporating spatial dispersion and wave-breaking effects. At moderate intensities (\ensuremath{\sim}${10}^{14}$ W/${\mathrm{cm}}^{2}$), the nonlinearity has little effect on the calculated reflectivity, but at higher intensities the reflectivity is greater than that predicted by a linear model. The discrepancy is small for a steplike plasma (with a density gradient scale length L\ensuremath{\sim}0.01${\ensuremath{\lambda}}_{0}$), but increases for longer scale lengths (L\ensuremath{\sim}${\ensuremath{\lambda}}_{0}$). In the case of p-polarized light, the angular dependence of reflectivity is sharpened at higher intensities, and resonance effects play an important role even for very short scale lengths.

Experimental studies of stimulated Raman scattering in reactor-size, laser-produced plasmas

In this paper the results of experiments designed to provide information regarding the scaling of the stimulated Raman scattering (SRS) instability in long-scale-length, inertial- confinement-fusion-reactor-size plasmas are described. Reactor-scale plasma conditions were experimentally simulated by exploding thin CH foils with nine beams of the Nova laser [Rev. Sci. Instrum. 57, 2101 (1986)]. Using a one-dimensional hydro model, the target and irradiation parameters were chosen to produce the largest-possible scale length plasma consistent with the energy and pulse width capabilities of Nova. The SRS emissions, driven by both the nine plasma-production beams and a tightly focused and delayed tenth beam, were recorded using an angularly resolved photodiode array and a temporally resolved optical spectrometer. These measurements verified the production of a reactor-size plasma (L/λ ≊ 5000) and indicated that the absolute levels of SRS backscatter detected in the present experiments have not increased relative to those measured in previous experiments despite a calculated sevenfold increase in the plasma density-gradient scale length. Inverse bremsstrahlung absorption of the scattered light is believed to be the dominant mechanism for this apparent saturation and, in this parameter regime, may enhance the effective thermal coupling between the incident laser and plasma densities below 0.15ncrit where absorption of the incident laser light is typically weak.

Three-wave interactions and type II irregularities in the equatorial electrojet

A spectral description of a model of convective turbulence in the E region of the equatorial ionosphere is considered. It is demonstrated that three complex modes are sufficient to generate a stochastic steady state, in a manner analogous to similar models of the Euler equation and of drift waves. Complex terms in the nonlinear coupling coefficient are retained that are usually dropped on the basis of large kL, where L is the density gradient scale length. In addition, some parallel damping is included. Parameters that lead to a stochastic steady state are then found by explicitly solving for the wave vectors that reproduce growth rates and nonlinear coupling terms from a known stochastic case of a drift wave model.

A phenomenological model for cross-field plasma transport in non-ambipolar scrape-off layers

A simplified two-fluid transport model which includes phenomenological coefficients of particle diffusion, mobility, and thermal diffusivity is used to investigate the effects of nonambipolar particle transport on scrape-off layer (SOL) plasma profiles. A computer code (BSOLRAD3) has been written to iteratively solve for 2-D cross-field plasma density, potential, and electron temperature profiles for arbitrary boundary conditions, including segments of “limiters” that are electrically conducting or non-conducting. Numerical results are presented for two test cases: (1) a 1-D slab geometry showing the interdependency of the density, potential, and temperature gradient scale lengths on particle diffusion, mobility, and thermal diffusivity coefficients and limiter bias conditions, and (2) a 2-D geometry illustrating E × B plasma flow effects. It is shown that the SOL profiles can be quite sensitive to non-ambipolarity conditions imposed by the limiter and, in particular, whether the limiter surfaces are biased. Such effects, if overlooked in SOL transport analysis, can lead to erroneous conclusions about the magnitude of the local ambipolar diffusion coefficient.

Intense turbulence in the polar mesosphere : rocket and radar measurements

Simultaneous observations of polar mesospheric summer echoes (PMSE) have been made with two different frequency radars during the launch of a sounding rocket designed to measure the fluctuations in the electron density in the same height range. The cross-section for radar backscatter deduced from the rocket probe data under the assumption of isotropic turbulence is in reasonable agreement with the measured signals at both 53.5 MHz with the mobile SOUSY radar and 224 MHz with the EISCAT VHF radar, which correspond to backscatter wavelengths of about 3 and 0.75 m, respectively. Some controversy exists over the relative roles of turbulent scatter vs specular reflections in PMSE. A number of characteristics of the data obtained in this experiment are consistent with nearly isotropic, intense meter-scale turbulence on this particular day. Since equally compelling arguments for the importance of an anisotropic-type mechanism have been presented by other experimenters studying PMSE, we conclude that both isotropic and anisotropic mechanisms must operate. We have found the inner scale for the electron fluctuation spectrum, which corresponds to the diffusive subrange for that fluid, and have compared it to the inner scale for the neutral gas. The latter was found from the Kolmogorov microscale, which in turn depends on the energy dissipation rate in the gas. We found the dissipation rate from the spectral width of the 53.5 MHz backscatter signal and from the rocket electron density fluctuation data. The diffusive subrange was found to occur at a wavelength a factor of about 10 times smaller than the viscous subrange. This corresponds to a Schmidt number of about 100. High Schmidt numbers have been reported in recent measurements of the diffusion coefficient of the electrons in this height range made with the EISCAT incoherent scatter radar. About 15 min after the rocket flight an extremely high radar reflectivity was found with the SOUSY system. We have been able to reproduce this high level theoretically by scaling the rocket data with an increase in the neutral turbulence energy dissipation rate by a factor of 14 as deduced from the SOUSY spectral width, an increase in the electron density which is consistent with riometer data, and a 33% decrease in the electron density gradient scale length which is hypothesized. We also estimate the radar reflectivity at 933 MHz and conclude that signals in excess of thermal scatter levels would have occurred at the peak of the event studied, provided that the electron fluctuation spectrum decreases as k −7 in the viscous subrange. If the spectrum has an exponential form, however, a turbulent source cannot explain the enhanced 933 MHz echoes reported by EISCAT.

Nonlinear evolution of the unmagnetized ion Rayleigh–Taylor instability

The nonlinear evolution of the unmagnetized ion Rayleigh–Taylor instability is investigated. A nonlinear state corresponding to localized clumps of high-density plasma is obtained analytically. The characteristic scale size of the clumps is given by Δ∼D(gLn)−1/2, where g is the gravitational acceleration, Ln is the density gradient scale length, and D is a diffusion coefficient associated with the short-scale dissipation processes in the system. It is shown numerically that this nonlinear state may be both accessible and stable.

Backscattered light near the incident laser wavelength from 0.35 μm irradiated long scale length plasmas

Backscattered light at wavelengths near the incident laser wavelength of 0.35 μm irradiated thin targets has been spectrally and temporally resolved. Low-Z targets show significant backscattered signals even after the foils burn through when the density gradient scale length is approximately 1000 times the laser wavelength. The results show the importance of considering seeding mechanisms in the plasma.

Efficiency of caviton formation as a function of plasma density gradient

The effect of a zeroth-order density gradient on the development of cavitons has been investigated experimentally and numerically. The cavitons were produced via excitation of electron plasma waves (EPW) with a modest (E2/4πnTe≪1) resonant radio-frequency pump. The location of the resonance, on an inverse-parabolic density profile, was varied, with all other parameters being held constant. The depth of the caviton, and the strength of its associated trapped electric fields, are found to depend strongly on the density gradient scale length at the critical layer, with a maximum occurring when this length is infinite, at the flat top of the density profile. The results are accounted for by the dependence on the density gradient of the EPW convection rate and wave-breaking time. The study helps illuminate recent large-scale ionospheric density modification experiments.

Theory of ion temperature gradient instabilities: Thresholds and transport

An investigation of the ion temperature gradient instability that focuses on the behavior of the mode when it is weakly unstable is presented. It is shown that ηi= (2)/(3) is the threshold for long wavelength instability, ηi being the ratio of the density gradient scale length to the ion temperature gradient scale length. For ηi>0.902, a short wavelength mode is concomitantly unstable. The transport resulting from both the short and the long wavelengths is shown to constitute a ‘‘soft’’ turn on of anomalous transport for ηi<2, with long (short) wavelengths dominating for high (weak) enough collisionality. The point ηi=2 represents a ‘‘hard’’ threshold for transport: Beyond this point, large, collisionless transport over all wavelengths is precipitated. A nonlocal collisional theory as well as a kinetic theory with a Krook collision operator are presented to describe the progression from weak to strong instability. Transport estimates, based on mixing length arguments, are given for the various regimes incorporated by the critical points ηi= (2)/(3) , 0.902, and 2.

Hydrodynamic stability and the direct drive approach to laser fusion

In order for inertial confinement fusion (ICF) capsules to achieve high gain, they must not break up because of hydrodynamic instability, the fuel should be maintained on a low adiabat, and the peak laser intensities should be below thresholds of various plasma instabilities. In this paper the results of two-dimensional computer calculations modeling the growth of hydrodynamic instabilities of slabs directly driven with a laser are reported. In order for direct drive capsules to succeed with capsule finishes of 100–300 Å, the capsules must be imploded on an adiabat a factor of 3– 4 in pressure above a Fermi-degenerate adiabat or techniques must be developed for producing density gradient scale lengths of about half the shell thickness. These conclusions depend only weakly on the laser intensity.

Absorption of subpicosecond ultraviolet laser pulses in high-density plasma

The absorption of 250 fs KrF laser pulses incident on solid targets of aluminum, copper and gold has been measured for normal incidence as a function of laser intensity in the range of 1012–1014 W cm−2 and as a function of polarization and angle of incidence for the intensity range of 1014−2.5×1015 W cm−2. As the intensity increases from 1012 W cm−2 the reflectivity at normal incidence changes from the low-intensity mirror reflectivity value to values in the range of 0.5–0.61 at 1014 W cm−2. For this intensity maximum absorption of 63–80% has been observed for p-polarized radiation at angles of incidence in the range of 54°–57°, increasing with atomic number. The results are compared with the expected Fresnel reflectivity from a sharp vacuum-plasma interface with the refractive index given by the Drude model and also to numerical calculations of reflectivity for various scale length density profiles. Qualitative agreement is found with the Fresnel/Drude model and quantitative agreement is noticed with the numerical calculations of absorption on a steep density profile with normalized collision frequencies, v/ω, in the range of 0.13–0.15 at critical density and normalized density gradient scale lengths, L/λ0, in the range of 0.018–0.053 for a laser intensity of 1014 W cm−2.

Comment on ‘‘Energy and nonlinearity considerations for the enhanced plasma wave model of Raman scattering’’ [Phys. Fluids B 1, 1073 (1989)

The interpretation of some recent experiments [Phys. Rev. Lett. 60, 1018 (1988); Phys. Fluids 30, 1795 (1988)] by Simon and Short [Phys. Fluids B 1, 1073 (1989), referred to as SS4] is discussed. The predicted angular distribution is almost narrow enough to correspond to the data discussed in SS4, but not to correspond to data from previous experiments having similar plasma-wave damping but much smaller density-gradient scale length. Examination of the time dependence of the Raman spectrum in all the data, rather than only the single case discussed in SS4, shows that the hydrodynamic behavior, rather than the vanishing of quarter critical, determines the time of maximum emission. Finally, it is notable that the analysis of the power flow in SS4 requires that most of the laser energy be converted into intense electron beams with narrow velocity distributions having intensities of 20× the average laser power. Though not impossible, this seems unlikely.

The stability of a compressible stratified shear layer

The stability of a stratified shear layer is studied using the compressible magnetohydrodynamic (MHD) equations, including an effective gravity term to represent curvature effects of the flow and magnetic field line geometry. A general eigenvalue equation is derived for a two-dimensional MHD fluid. For the case of a hyperbolic tangent shear flow and exponential density profile, it is found that in the Boussinesq approximation the compressibility raises the critical Richardson number from (1)/(4) to as much as (1)/(2) , with the exact value depending on the value of the magnetic field at infinity. Under the approximation of a strong asymptotic magnetic field, but without invoking the Boussinesq approximation, it is shown both analytically and numerically that the density gradient terms cause the shear instability to be dispersive. In particular, the long-wavelength stability boundary for the case of Richardson number J=0 is characterized by a normalized phase velocity c=−1. This result implies that the solution for the stability boundary found by Satyanarayana, Lee, and Huba [Phys. Fluids 30, 81 (1987)], which required that c=−β/2 (where β is the ratio of the velocity and density gradient scale lengths), is not valid, in general.

X-ray emission caused by Raman scattering in long-scale-length plasmas.

By analysis of data from a specific set of laser-plasma interaction experiments, it is argued that stimulated Raman scattering (SRS) is responsible for the hard (>30-keV) x rays emitted from the targets. The Novette laser [K. R. Manes et al., Laser Part. Beams 3, 173 (1985)] was used to irradiate thick, gold targets with up to 4 kJ of 0.53-\ensuremath{\mu}m light in 1-ns, Gaussian pulses at average intensities of (1--200)\ifmmode\times\else\texttimes\fi{}${10}^{14}$ W/${\mathrm{cm}}^{2}$, producing approximately planar plasmas with temperatures of order 3 keV and density-gradient scale lengths of order 250 \ensuremath{\mu}m. The spectrum, amplitude, and timing of the hard x rays emitted by the plasma were measured along with various properties of the scattered light. All the data are consistent with the hypothesis that SRS is the source of the hot electrons that emit the x rays, and some data conflict with any other plausible hypothesis.

Edge fluctuation measurements in textor with a thermal neutral beam probe

Simultaneous measurements of radial profiles of the electron density and its fluctuations in the TEXTOR scrape-off layer have been performed with a thermal neutral lithiam beam probe. Fluctuation levels and frequency spectra are compared with each other for different plasma states, e.g. helium and hydrogen plasmas, detatched and attached plasmas. The influence of the pump limiter is also investigated. In all plasma states, broadband fluctuations with a similar frequency spectrum are observed. The fluctuation levels are well correlated with density gradient scale lengths and with line-averaged electron densities. The different features between the various are also presented.

Nonlocal theory of the Rayleigh–Taylor instability in the limit of unmagnetized ions

A nonlocal theory of the Rayleigh-Taylor instability is developed that applies to both magnetized and unmagnetized ion systems. The distinction between these two regimes is the time scale of the instability: for γ<Ωi the ions can be considered to be magnetized, and for γ>Ωi the ions can be considered to be unmagnetized, where γ is the growth rate and Ωi is the ion gyrofrequency. Both analytical and numerical results are presented that contrast the behavior of the Rayleigh-Taylor instability in these two regimes. In the long wavelength limit (kyLn≪1, where ky is the wavenumber and Ln is the density gradient scale length), it is found that the growth rate of the Rayleigh-Taylor instability in the unmagnetized ion limit is γ≂(kyg/A)1/2, where A is the Atwood number; this is quite different from the growth rate in the magnetized ion limit which is γ=(kygA)1/2. Another distinction between the two limits is that in the unmagnetized ion regime the fastest growing mode is not the lowest-order mode; again this is different from the magnetized ion regime in which the fastest growing mode is the lowest order mode. Finally, these results are applied to recent experimental observations (e.g., AMPTE [J. Geophys. Res. 92, 5777 (1987)], NRL laser experiment [Phys. Rev. Lett. 59, 2299 (1987)]).

Nonlocal theory of long‐wavelength plasma waves associated with sporadic E layers

In this paper we calculate the nonlocal growth rate of gradient drift plasma waves under conditions where the electron density gradient scale length changes with altitude. The results are compared with the local growth rate and discussed in the context of the kilometer‐scale waves which have been observed in the vicinity of mid‐latitude sporadic E layers. These large‐scale waves drastically violate the local approximation, kLm»1, where k is the irregularity wave number and Lm is the minimum gradient scale length on the edge of a layer. The first step in the analysis is to derive a general eigenmode equation, starting with the same assumptions usually used in the derivation of the local dispersion relation for long wavelength waves. Modeling a sporadic E layer as a slab, the nonlocal growth rate spectrum is found by solving the eigenmode equation for this profile. The solution is an algebraic dispersion relation with a growth rate spectrum which is roughly proportional to k, rather than the k2 dependence predicted by conventional local theory at long wavelengths. At short wavelengths the nonlocal growth rate determined with the slab is unbounded, in disagreement with local theory. The slab is an inadequate model for short wavelength waves and a bound on the growth rate is instead derived from a theory which can be applied to any realistic profile with nonzero Lm. At short wavelengths this bound is identical to the local growth rate expression, while at long wavelengths the bound remains proportional to k and thus is consistent with the dispersion relation for a slab. Nonlocal effects alone do not explain the dominance of kilometer scales, but they do tend to favor the excitation of long wavelengths.

Entropy gradient-driven electron inertial instabilities

In the vicinity of a density transition region or surface, electron temperature fluctuations are known to drive low frequency electromagnetic surface waves which may be connected with the self-generation of magnetic fields in the laser–plasma interaction. Two methods of destabilization are considered. In the first approach an accelerating plasma surface is considered and this leads to instability only when an accompanying temperature gradient scale length is at most as large as the density gradient scale length. In the second approach the introduction of current is also found to cause instability near the plasma surface provided an identical temperature gradient threshold is exceeded. In both cases the growth rates for each instability are enhanced by the square root of the ion to electron mass ratio over the associated growth rates for ion-inertial-type instability.

Ion temperature gradient mode in the weak density gradient limit

The toroidal ion temperature gradient mode (ITG mode) is analyzed in the fluid limit for flat density profiles, typical of many H-mode plasmas. It is shown that the stability boundary is not Ln/LTi=const ≊O(1 ■2) (Ln and LTi are the density and ion temperature gradient scale lengths, respectively), as typically quoted. Rather, for large Ln/R the stability threshold is characterized by LTi/R =const≊O(0.1) with R the major radius. This result could substantially reduce the impact of ITG-mode turbulence, allowing a reconciliation of present drift wave transport models with improved H-mode energy confinement.